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G = C5×Dic5order 100 = 22·52

Direct product of C5 and Dic5

direct product, metacyclic, supersoluble, monomial, A-group

Aliases: C5×Dic5, C52C20, C10.C10, C525C4, C10.4D5, C2.(C5×D5), (C5×C10).1C2, SmallGroup(100,6)

Series: Derived Chief Lower central Upper central

C1C5 — C5×Dic5
C1C5C10C5×C10 — C5×Dic5
C5 — C5×Dic5
C1C10

Generators and relations for C5×Dic5
 G = < a,b,c | a5=b10=1, c2=b5, ab=ba, ac=ca, cbc-1=b-1 >

2C5
2C5
5C4
2C10
2C10
5C20

Permutation representations of C5×Dic5
On 20 points - transitive group 20T25
Generators in S20
(1 5 9 3 7)(2 6 10 4 8)(11 17 13 19 15)(12 18 14 20 16)
(1 2 3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)
(1 16 6 11)(2 15 7 20)(3 14 8 19)(4 13 9 18)(5 12 10 17)

G:=sub<Sym(20)| (1,5,9,3,7)(2,6,10,4,8)(11,17,13,19,15)(12,18,14,20,16), (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20), (1,16,6,11)(2,15,7,20)(3,14,8,19)(4,13,9,18)(5,12,10,17)>;

G:=Group( (1,5,9,3,7)(2,6,10,4,8)(11,17,13,19,15)(12,18,14,20,16), (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20), (1,16,6,11)(2,15,7,20)(3,14,8,19)(4,13,9,18)(5,12,10,17) );

G=PermutationGroup([(1,5,9,3,7),(2,6,10,4,8),(11,17,13,19,15),(12,18,14,20,16)], [(1,2,3,4,5,6,7,8,9,10),(11,12,13,14,15,16,17,18,19,20)], [(1,16,6,11),(2,15,7,20),(3,14,8,19),(4,13,9,18),(5,12,10,17)])

G:=TransitiveGroup(20,25);

C5×Dic5 is a maximal subgroup of
C523C8  Dic52D5  C5⋊D20  C522Q8  D5×C20  C522Dic3  He55C4  C50.C10  He56C4
C5×Dic5 is a maximal quotient of
He55C4  C50.C10

40 conjugacy classes

class 1  2 4A4B5A5B5C5D5E···5N10A10B10C10D10E···10N20A···20H
order124455555···51010101010···1020···20
size115511112···211112···25···5

40 irreducible representations

dim1111112222
type+++-
imageC1C2C4C5C10C20D5Dic5C5×D5C5×Dic5
kernelC5×Dic5C5×C10C52Dic5C10C5C10C5C2C1
# reps1124482288

Matrix representation of C5×Dic5 in GL2(𝔽11) generated by

90
09
,
80
07
,
010
10
G:=sub<GL(2,GF(11))| [9,0,0,9],[8,0,0,7],[0,1,10,0] >;

C5×Dic5 in GAP, Magma, Sage, TeX

C_5\times {\rm Dic}_5
% in TeX

G:=Group("C5xDic5");
// GroupNames label

G:=SmallGroup(100,6);
// by ID

G=gap.SmallGroup(100,6);
# by ID

G:=PCGroup([4,-2,-5,-2,-5,40,1283]);
// Polycyclic

G:=Group<a,b,c|a^5=b^10=1,c^2=b^5,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

Export

Subgroup lattice of C5×Dic5 in TeX

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